A global optimization algorithm based on multi-loop neural network control

doi:10.21629/JSEE.2019.05.17

Journal of Systems Engineering and Electronics ›› 2019, Vol. 30 ›› Issue (5): 1007-1024.doi: 10.21629/JSEE.2019.05.17

收稿日期:2018-12-13 出版日期:2019-10-08 发布日期:2019-10-09

A global optimization algorithm based on multi-loop neural network control

Baiquan LU(), Chenlong NI*(), Zhongwei ZHENG(), Tingzhang LIU()

School of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200444, China

Received:2018-12-13 Online:2019-10-08 Published:2019-10-09
Contact: Chenlong NI E-mail:lbq123188@aliyun.com;chalone0808@icloud.com;zw.zheng2@gmail.com;liutzh@staff.shu.edu.cn
About author:LU Baiquan was born in 1963. He received his Ph.D. degree in thermal engineering from Tsinghua University in 1997. He is now an associate professor at School of Mechatronic Engineering and Automation, Shanghai University. His research interests include computational intelligence and nonlinear system control. E-mail: lbq123188@aliyun.com|NI Chenlong was born in 1995. He received his B.S. degree from Nanjing Institute of Technology in 2017. He is currently pursuing his M.S. degree at School of Mechatronic Engineering and Automation, Shanghai University. His major research interests include optimization, particle swarm optimization and deep learning. E-mail: chalone0808@icloud.com|ZHENG Zhongwei was born in 1993. He received his B.S. degree from Changchun University of Science and Technology in 2016. He is currently pursuing his M.S. degree at School of Mechatronic Engineering and Automation, Shanghai University. His major research interests include optimization, particle swarm optimization and deep learning. E-mail: zw.zheng2@gmail.com|LIU Tingzhang is a professor at School of Mechatronic Engineering and Automation, Shanghai University. He received his Ph.D. degree in mechanical engineering from Xi'an Jiaotong University in 1996. His research interests include modeling and control for complex system, energy-saving and control for building system. E-mail: liutzh@staff.shu.edu.cn
Supported by:
the National Natural Science Foundation of China(61273190);This work was supported by the National Natural Science Foundation of China (61273190)

摘要/Abstract

Abstract:

This paper proposes an optimization algorithm based on a multi-loop control system with a neural network controller, in which the objective function that is used is the control plant of each sub-control system. To obtain the global optimization solution from a control plant that has many local minimum points, a transformation function is presented. On the one hand, this approach changes a complex objective function into a simple function under the condition of an unchanged globally optimal solution, to find the global optimization solution more easily by using a multi-loop control system. On the other hand, a special neural network (in which the node function can be simply positioned locally) that is composed of multiple transformation functions is used as the controller, which reduces the possibility of falling into local minimum points. At the same time, a filled function is presented as a control law; it can jump out of a local minimum point and move to another local minimum point that has a smaller value of the objective function. Finally, 18 simulation examples are provided to show the effectiveness of the proposed method.

Key words: global optimization, neural networks, control system, transformation function, filled function method

. [J]. Journal of Systems Engineering and Electronics, 2019, 30(5): 1007-1024.

Baiquan LU, Chenlong NI, Zhongwei ZHENG, Tingzhang LIU. A global optimization algorithm based on multi-loop neural network control[J]. Journal of Systems Engineering and Electronics, 2019, 30(5): 1007-1024.

图/表 22

Function	Dimension	Feasible region	Global optimum value
$F_1 = \sum\limits_{i = 1}^n {(x_i^2-10\cos (2\mathtt{π} x_i)+10)}$	$n = 30$	$-5.12 {\leqslant} x_i {\leqslant} 5.12$	$F_1(0, 0, \ldots, 0) = 0$
$F_2 = -20\exp \left({-0.2\sqrt {\frac{1}{n}\sum\limits_{i = 1}^n {x_i^2 } }} \right)- \\\exp \left({\frac{1}{n}\sum\limits_{i = 1}^n {\cos (2\mathtt{π} x_i)} } \right)+20+\exp (1)$	$n = 30$	$-32 {\leqslant} x_i {\leqslant}32$	$F_2(0, 0, \ldots, 0) = 0$
$F_3 = \frac{1}{4 000}\sum\limits_{i = 1}^n {x_i^2 } - \prod\limits_{i = 1}^n {\cos \left({\frac{x_i }{\sqrt i }} \right)} + 1$	$n = 30$	$-600 {\leqslant} x_i {\leqslant} 600$	$F_3(0, 0, \ldots, 0) = 0$
$F_4 = \frac{\mathtt{π}}{n}\bigg\{10\sin^2(\mathtt{π} y_1)+(y_n-1)^2+\sum\limits^{n-1}_{i = 1}(y_i-1)^2\cdot\\$ $[1+10\sin^2(\mathtt{π} y_{i+1})]\bigg\}, y_i = 1+\frac{1}{4}(x_i+1)$	$n = 30$	$-50 {\leqslant} x_i {\leqslant} 50$	$F_4(-1, -1, \ldots, -1) = 0$
$F_5 = \frac{1}{10}\bigg\{ \sin ^2(3\mathtt{π} x_1)+\sum\limits_{i = 1}^{n-1} (x_i-1)^2 \cdot\\$ $[1+\sin ^2(3\mathtt{π} x_{i+1})]+\\$ $(x_n-1)^2[1+\sin ^2(2\mathtt{π} x_n)]\bigg\} $	$n = 30$	$-50 {\leqslant} x_i {\leqslant} 50$	$F_5(1, 1, \ldots, 1) = 0$
$F_6 = \frac{1}{n}\sum\limits_{i = 1}^n (x_i^4-16x_i^2+5x_i)$	$n = 30$	$-5 {\leqslant} x_i {\leqslant} 5$	$\begin{array}{l} F_6(-2.903 53, \ldots, -2.903 53) =-78.332 3\end{array}$
$F_7 = \sum\limits_{i = 1}^{n-1} [100(x_i^2-x_{i+1})^2+(x_i-1)^2] $	$n = 30$	$-5 {\leqslant} x_i{\leqslant} 10$	$F_7(1, 1, \ldots, 1) = 0$
$F_8 = \sum\limits_{i = 1}^n {x_i^2 } $	$n = 30$	$-100 {\leqslant} x_i {\leqslant}100$	$F_8(0, 0, \ldots, 0) = 0$
$F_9 = \sum\limits_{i = 1}^n {x_i^4 }$	$n = 30$	$-1.28 {\leqslant} x_i {\leqslant} 1.28$	$F_9(0, 0, \ldots, 0) = 0$
$F_{10} = \sum\limits_{i = 1}^n \left[\sum\limits_{j = 1}^i x_j\right]^2$	$n = 30$	$-100 {\leqslant} x_i {\leqslant} 100$	$F_{10}(0, 0, \ldots, 0) = 0$
$F_{11} = - \sum\limits_{i = 1}^4 {c_i } \exp \bigg\{ - \sum\limits_{j = 1}^6 a_{i, j}(x_j-p_{i, j})^2 \bigg\}\\$ $\boldsymbol{p}_{4 \times 6}=\left[\begin{array}{cccccc} 0.1312 & 0.1696 & 0.5569 & 0.0124 & 0.8283 & 0.5886 \\ 0.2329 & 0.4135 & 0.8307 & 0.3736 & 0.1004 & 0.9991 \\ 0.2348 & 0.1451 & 0.3522 & 0.2883 & 0.3047 & 0.6650 \\ 0.4047 & 0.8828 & 0.8732 & 0.5743 & 0.1091 & 0.0381 \end{array}\right]\\$$\boldsymbol{a}_{4 \times 6}=\left[\begin{array}{cccccc}10 & 3 & 17 & 3.5 & 1.7 & 8 \\ 0.05 & 10 & 17 & 0.1 & 8 & 14 \\ 3 & 3.5 & 1.7 & 10 & 17 & 8 \\ 17 & 8 & 0.05 & 10 & 0.1 & 14\end{array}\right]$$\boldsymbol{c}_{1} \times 4=\left[\begin{array}{llll} 1.0 & 1.2 & 3.0 & 3.2 \end{array}\right]$	$n = 6$	$0 {\leqslant} x_i {\leqslant} 1$	$ F_{11}(0.201 7, 0.150 0, 0.476 9, 0.275 3, 0.311 7, 0.657 3) =-3.322 4$
$F_{12} = \left\{\sum\limits_{i = 1}^5 {i\cos [(i+1)x_1+i} \right\}\left\{\sum\limits_{i = 1}^5 {i\cos [(i+1)x_2+i} \right\}$	$n = 2$	$-10 {\leqslant} x_i {\leqslant}10$	$\begin{array}{l} F_{12}(-0.199 678 75, -5.858 056 86) =\\F_{12}(0.425 128 4, -0.199 678 87) =\\F_{12}(4.858 056 87, 5.482 864 36) =\\F_{12}(5.482 864 36, 4.858 056 87) =\\-186.073 090 883 097 1\end{array}$
$F_{13} = - \sum\limits_{i = 1}^5 {[(x_1-s_{i, 1})^2}+(x_2- s_{i, 2})^2+(x_3-s_{i, 3})^2+$$(x_4-s_{i, 4})^2+c_i]^{-1}$$F_{14} = - \sum\limits_{i = 1}^7 {[(x_1-s_{i, 1})^2}+(x_2- s_{i, 2})^2+(x_3-s_{i, 3})^2+$$(x_4-s_{i, 4})^2+c_i]^{-1}$$F_{15} = - \sum\limits_{i = 1}^{10} {[(x_1-s_{i, 1})^2}+(x_2- s_{i, 2})^2+(x_3-s_{i, 3})^2+$$(x_4-s_{i, 4})^2+c_i]^{-1}$$\boldsymbol{c}=\left[\begin{array}{cccccccccc} 0.1 & 0.2 & 0.2 & 0.4 & 0.4 & 0.6 & 0.3 & 0.7 & 0.5 & 0.5 \end{array}\right]$$\boldsymbol{s}=\left[\begin{array}{cccccccccc} 4 & 1 & 8 & 6 & 3 & 2 & 5 & 8 & 6 & 7 \\ 4 & 1 & 8 & 6 & 7 & 9 & 5 & 1 & 2 & 3.6 \\ 4 & 1 & 8 & 6 & 3 & 2 & 3 & 8 & 6 & 7 \\ 4 & 1 & 8 & 6 & 7 & 9 & 3 & 1 & 2 & 3.6 \end{array}\right]^\text{T}$	$n = 4$	$0 {\leqslant} x_i {\leqslant} 10$	$ F_{13}(4, 4, 4, 4) = -10.153 2\\F_{14}(4, 4, 4, 4) = -10.402 9\\F_{15}(4, 4, 4, 4) = -10.536 4$
$F_{16} = \sum\limits_{i = 1}^n {(\sum\limits_{k = 0}^{k\max } {[a^k\cos (2\mathtt{π} b^k(x_i+0.5))])} }-\\n \sum\limits_{k = 0}^{k\max } {[a^k\cos (2\mathtt{π} b^k \cdot 0.5)]}, a = 0.5, b = 3, k\max = 20$	$n = 30$	$-0.5 {\leqslant} x_i {\leqslant} 0.5$	$F_{16}(0, 0, \ldots, 0) = 0$
$F_{17} = 418.982 9 n-\sum\limits_{i = 1}^n {x_i\sin (\sqrt {\|x_i\|})} $	$n = 30$	$-500 {\leqslant} x_i {\leqslant}500$	$F_{17}(420.97, 420.97, \ldots, 420.97) =0.000 381 827$

Function	Parameter of the filled function	${{{\mathit{\boldsymbol{k}}}}}_p / {{{\mathit{\boldsymbol{k}}}}}_I / {{{\mathit{\boldsymbol{k}}}}}_D /{{{\mathit{\boldsymbol{k}}}}}_0 / {{{\mathit{\boldsymbol{w}}}}}$ on [-0.5, 0.5]	$\eta _1 / \eta _2 /\eta _3$	$a_1$	$MN/ /TT/ FMN$
$F_{1}$	$d_1$ = 1e-8, $d_2$ = 1e-25, $d_3$ = 23, $d_4$ = 0.1, $d_5$ = 3	1e-3/1e-3/1e-3/1e-3/1e-3	1e-3/1e-5/1e-4	3.3e-4	10 000/3 000/100
$F_{2}$	$d_1$ = 1e-8, $d_2$ = 1e-25, $d_3$ = 40, $d_4$ = 0.2, $d_5$ = 28	0.01/0.01/0.01/0.01/0.01	1e-3/1e-1/1e-5	2e-4	50 000/6 000/100
$F_{3}$	$d_1$ = 1e-4, $d_2$ = 1e-15, $d_3$ = 30, $d_4$ = 1, $d_5$ = 11	1/1/1/1e-6/1	1.0/1e+2/1e-2	7e-6	10 000/2 500/80
$F_4$	$d_1$ = 1e-3, $d_2$ = 1e-17, $d_3$ = 12, $d_4$ = 0.3, $d_5$ = 16	0.01/0.01/0.01/0.01/0.01	1/1e-6/1e-5	1.2e-4	10 000/3 000/150
$F_{5}$	$d_1$ = 1e-3, $d_2$ = 1e-18, $d_3$ = 29, $d_4$ = 1, $d_5$ = 9	0.1/0.1/0.1/1e-5/0.1	2e-1/1e-2/1e-4	1e-4	10 000/3 000/80
$F_{6}$	$d_1$ = 1e-4, $d_2$ = 1e-9, $d_3$ = 5, $d_4$ = 1.05, $d_5$ = 9.5	0.01/0.01/0.01/1e-4/0.1	5e-2/1e-1/2e-4	1e-3	1 000/180/60
$F_{7}$	$d_1$ = 1e-9, $d_2$ = 1e-16, $d_3$ = 28, $d_4$ = 2, $d_5$ = 16	1e-2/1e-2/1e-2/1e-5/0.1	7e-4/1e-5/1e-4	2e-4	40 000/6 000/100
$F_{8}$	$d_1$ = 1e-26, $d_2$ = 1e-30, $d_3$ = 8, $d_4$ = 1, $d_5$ = 6	1e-2/1e-2/1e-2/1e-4/1e-2	9e-3/1e-5/1e-2	1e-3	10 000/120/60
$F_{9}$	$d_1$ = 1e-3, $d_2$ = 1e-10, $d_3$ = 15, $d_4$ = 1, $d_5$ = 9	1e-2/1e-2/1e-2/1e-4/1e-2	1e-2/1e-2/1e-3	1e-4	6 000/2 000/60
$F_{10}$	$d_1$ = 1e-6, $d_2$ = 1e-18, $d_3$ = 26, $d_4$ = 1, $d_5$ = 13	0.1/0.1/0.1/0.01/0.1	9e-5/1e-4/1e-4	1e-4	20 000/4 000/60
$F_{11}$	$d_1$ = 1e-6, $d_2$ = 1e-6, $d_3$ = 11, $d_4$ = 1, $d_5$ = 7	0.1/0.1/0.1/0.1/0.1	9e-2/1e-3/1e-4	2e-1	1 500/200/60
$F_{12}$	$d_1$ = 1e-4, $d_2$ = 1e-7, $d_3$ = 6.6, $d_4$ = 5, $d_5$ = 8	0.1/0.1/0.1/0.1/0.1	1e-3/0.01/1e-4	1e-2	140/500/60
$F_{13}$	$d_1$ = 1e-5, $d_2$ = 1e-7, $d_3$ = 7, $d_4$ = 60, $d_5$ = 12	0.1/0.1/0.1/0.1/0.1	5.8/1e-4/1e-4	1e-3	3 000/1 000/100
$F_{14}$	$d_1$ = 7.2e-10, $d_2$ = 1e-9, $d_3$ = 5, $d_4$ = 20, $d_5$ = 15	0.1/0.1/0.1/0.1/0.1	8.8e-4/8.5e-4/9e-4	5e-3	50 000/1 500/250
$F_{15}$	$d_1$ = 7.2e-10, $d_2$ = 1e-9, $d_3$ = 5, $d_4$ = 21, $d_5$ = 15	0.1/0.1/0.1/0.1/0.1	9.5e-4/8.8e-4/5e-4	5e-3	35 000/1 800/300
$F_{16}$	$d_1$ = 2e-9, $d_2$ = 1e-19, $d_3$ = 23, $d_4$ = 1, $d_5$ = 14	0.01/0.01/0.01/1e-4/0.01	1e-5/1e-5/1e-5	1e-5	20 000/2 000/350
$F_{17}$	$d_1$ = 1e-8, $d_2$ = 1e-1, $d_3$ = 8, $d_4$ = 1, $d_5$ = 8	0.1/0.1/0.1/0.1/0.1	1e-5/1e-5/2e-5	3e-4	10 000/600/250
$F_{18}$	$d_1$ = 1e-15, $d_2$ = 1e-3, $d_3$ = 6, $d_4$ = 1, $d_5$ = 5	0.1/0.1/0.1/0.1/0.1	1e-5/4e-3/1e-3	5e-3	10 000/5 000/80

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Function	Average value	Best value	Worst value	Confidence interval	$N$/30	CPU time/s	$SA$
$F_{1}$	0	0	0	0$\pm $0	30/30	33.3	1e-15
$F_{2}$	0	0	0	0$\pm $0	30/30	82.1	1e-15
$F_{3}$	1.48e-17	0	3.33e-16	1.48e-17$\pm $1.44e-33	30/30	15.5	1e-15
$F_4$	5.66e-11	1.33e-12	3.02e-10	5.66e-11$\pm $4.8e-21	30/30	27.5	1e-9
$F_{5}$	5.30e-18	1.50e-19	2.60e-17	5.30e-18$\pm $1.62e-35	30/30	32.5	1e-15
$F_{6}$	-78.332 331 41	-78.332 331 41	-78.332 331 41	-78.332 331 41$\pm$2.99e-28	30/30	5.83	1e-8
$F_{7}$	3.75e-17	1.50e-19	3.92e-16	3.75e-17$\pm $3.1e-33	30/30	78.6	1e-15
$F_{8}$	0	0	0	0$\pm $0	30/30	5.1	1e-15
$F_{9}$	1.11e-24	6.36e-26	5.26e-24	1.11e-24$\pm $0.47e-48	30/30	9.7	1e-15
$F_{10}$	1.26e-16	1.73e-21	1.67e-15	1.26e-16$\pm $0.41e-31	30/30	156.6	1e-15
$F_{11}$	-3.322 368 011	-3.322 368 011	-3.322 368 011	-3.322 368 011$\pm$2.28e-23	30/30	0.18	1e-9
$F_{12}$	-186.730 908 8	-186.730 908 8	-186.730 908 8	-186.730 908 8$\pm$2.99e-28	30/30	0.98	1e-7
$F_{13}$	-10.153 199 68	-10.153 199 68	-10.153 199 68	-10.153 199 68$\pm$0.73e-22	30/30	1.18	1e-8
$F_{14}$	-10.402 940 57	-10.402 940 57	-10.402 940 57	-10.402 940 57$\pm$1.28e-24	30/30	11.58	1e-8
$F_{15}$	-10.536 409 82	-10.536 409 82	-10.536 409 82	-10.536 409 82$\pm$0.4e-25	30/30	27.4	1e-9
$F_{16}$	0	0	0	0$\pm $0	30/30	1 668	1e-15
$F_{17}$	0.000 381 827	0.000 381 827	0.000 381 83	0.000 381 827$\pm $0.979e-38	30/30	10.34	1e-8

Spring diameter	$D$
Spring wire diameter	$d$
Spring total number of turns	$n$
Rotation ratio	$c(c = D/d)$
Spring terminal type factor	$\lambda = 0.5$
Proportion of spring material/(N/mm$^3$)	$\rho = 7.8\times 10^{ - 5}$
Shear modulus of the spring material/(N/mm$^2$)	$G = 8.1\times 10^4$
Shearing stress of the spring wire/(N/mm$^2$)	$[\tau] = 444$
Spring support laps	$n_2 = 1.75$
Working load/N	$P = 700$

Algorithm	FEP	OGA/Q	CMA-ES	JADE	OLPSO-L	GPSO	Fsolve*	LargeScale*	GradObj*	The proposed
$F_{1}$	4.6e-2 $\pm $1.2e-2	0$\pm $0	1.76e+2$\pm $13.89	0$\pm $0	0$\pm $0	0$\pm $0	553$\pm $705	259$\pm $1 171	517$\pm $1 145	0$\pm $0
Rank	2	1	3	1	1	1	6	4	5	1
$F_{2}$	1.8e-2$\pm $2.1e-3	4.4e-16$\pm $4e-17	12.124$\pm $9.28	4.4e-15$\pm $0	4.14e-15$\pm $0	4.4e-015$\pm $2.4e-16	212$\pm $0.024	86$\pm $1 619	19.5$\pm $0.01	0$\pm $0
Rank	6	2	7	4	3	5	10	9	8	1
$F_{3}$	1.6e-2$\pm $2.2e-2	0$\pm $0	9.59e-16$\pm $3.5e-16	2.e-4$\pm $1.4e-3	0$\pm $0	0.003 7$\pm $0.001 2	0$\pm $0	4.87E-13$\pm $1.7E-25	0$\pm $0	1.48e-17$\pm $1.4e-33
Rank	6	1	3	5	1	5	1	4	1	2
$F_{7}$	5.06$\pm $5.87	0.75$\pm $0.11	2.33e-15$\pm $7.7e-16	0.32$\pm $1.1	1.26$\pm $1.4	0.042 0$\pm $0.003 5	618$\pm $407 070	6.64E-01$\pm $0.82	0.53$\pm $0.68	3.75e-17$\pm $3.1e-33
Rank	9	7	2	4	8	3	10	6	5	1
$F_{8}$	5.7e-4$\pm $1.3e-4	0$\pm $0	4.56e-16$\pm $1.13e-16	1.3e-54$\pm $9.2e-54	1.1e-38$\pm $1.3e-38	1.8326e-52$\pm $6.96e-53	0$\pm $0	2.4E-35$\pm $8.5E-36	1E-88$\pm $3.7E-89	0$\pm $0
Rank	8	1	7	3	5	4	1	6	2	1
$F_{17}$	14.98$\pm $52.6	3.03e-2$\pm $6.4e-4	3.15e+3$\pm $5.79e+2	7.1$\pm $28	3.8e-4$\pm $0	3.9e-004$\pm $0	12 700$\pm $856 347	4.95E+03$\pm $257 038	5.15E+03$\pm $322 436	0.000 381 827$\pm $0.98e-38
Rank	6	4	7	5	3	2	10	8	9	1
Average rank	6.17	2.83	4.83	3.67	3.5	3.3	6.3	6.17	5	1.17
Final rank	8	2	6	5	4	3	9	8	7	1

A global optimization algorithm based on multi-loop neural network control

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